I am unclear on the how to write the likelihood function in order to find the maximum likelihood estimate of $\theta$ from the following probability mass function (PMF), if I have 5 independent observations $(3,0,2,1,3)$.
$$P(X\vert\theta)=\begin{cases}{\frac{2\theta}{3}}& x=0 \\ \frac{\theta}{3}& x=1\\ \frac{1-\theta}{3} &x=2 \\ \frac{2(1-\theta)}{3} & x=3 \end{cases}$$
Now it is my understanding that the likelihood function is the product (in this case five times) of the PMF. Since each $i$ observation is different I have to write the corresponding function and multiply it by the next in the following way:
$$L(\theta|X)=\frac{2(1-\theta)}{3} \cdot \frac{2\theta}{3} \cdot \frac{1-\theta}{3} \cdot \frac{\theta}{3} \cdot \frac{2(1-\theta)}{3}$$
This seems right but I have the doubt (unfounded and irrational maybe?) that it should also include the realized values, like this:
$$L(\theta|X)=(3)\frac{2(1-\theta)}{3} \cdot (0)\frac{2\theta}{3} \cdot (2)\frac{1-\theta}{3} \cdot (1)\frac{\theta}{3} \cdot (3)\frac{2(1-\theta)}{3}$$
I hope I'm correct in using the first likelihood function, but if it is the second one could you please explain why?
Best Answer
Solved in comments, copied here:
You should only be multiplying the probabilities together and not the realized values. You may be confusing the concept of an expectation with the likelihood function, but the two are unrelated. – dsaxton